The Heat Kernel Method

Abstract

The goal of the heat kernel method is to express (2.40) as an integral over the fixed point set Mγ in M of the transformation γ. Here Mγ = M if γ is the identity. The method is based on the following observations about arbitrary elliptic differential operators D, acting on sections of a smooth vector bundle F over a compact manifold M, which admits a a direct sum decomposition F = F+ ⊕ F−. In our case, F = E ⊗ L, with the splitting F± = E± ⊗ L, and D is the spin-c Dirac operator. For the required facts about trace class operators, see for instance Hörmander [42, Sec. 19.1], or Duistermaat [19].